K-purity and orthogonality

Michel Hebert

Adamek and Sousa recently solved the problem of characterizing
the subcategories K of a locally $\lambda$-presentable category
C which are $\lambda$-orthogonal in C, using their concept
of K$\lambda$-pure morphism. We strengthen the latter
definition, in order to obtain a characterization of the classes
defined by orthogonality with respect to $\lambda$-presentable
morphisms (where $f : A \rightarrow B is called
$\lambda$-presentable if it is a $\lambda$-presentable object of
the comma category A/C). Those classes
are natural examples of reflective subcategories defined by proper
classes of morphisms. Adamek and Sousa's result follows from
ours. We also prove that $\lambda$-presentable morphisms are
precisely the pushouts of morphisms between $\lambda$-presentable
objects of C.